Problem: A circle with radius $9$ has a sector with a $120^\circ$ central angle. What is the area of the sector? ${81\pi}$ $\color{#9D38BD}{120^\circ}$ ${27\pi}$ ${9}$
Answer: First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. $A_c = \pi r^2$ $A_c = \pi (9)^2$ $A_c = 81\pi$ The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{120^\circ}{360^\circ} = \dfrac{A_s}{81\pi}$ $\dfrac{1}{3} = \dfrac{A_s}{81\pi}$ $\dfrac{1}{3} \times 81\pi = A_s$ $27\pi = A_s$